Extremal functions for Hardy's inequality with weight
Let \(\Omega\) be a \(C^2\) bounded domain in \(\mathbb{R}^N\) and \(\delta(x) = \text{dist} (x,\partial \Omega)\). \textit{H. Brezis} and \textit{M. Marcus} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 25, 217-237 (1997)] investigated the quantity \[ J_{\lambda} = J_{\lambda} (p,q,\eta) = \inf_{u\in H^1_0(\Omega)} \frac{\int_{\Omega} p|\nabla u|^2 - \lambda\int_{\Omega} \eta (u/\delta)^2}{\int_{\Omega}q(u/\delta)^2}, \leqno(1) \] where \(\lambda\in\mathbb{R}\), \(p,q\in C^1(\overline{\Omega})\) and \(p,q > 0\) in \(\overline{\Omega}\), \(\eta\in C^0(\overline{\Omega})\), \(\eta > 0\) in \(\Omega\) and \(\eta=0\) on \(\partial\Omega\). Assuming that \(\max_{\partial\Omega} p/q = 1\), they proved that there exists a finite constant \(\lambda^* = \lambda^* (\Omega)\) such that \(J_{\lambda} = 1/4\) for all \(\lambda \leq \lambda^*\), and \(J_{\lambda} < 1/4\) for every \(\lambda > \lambda^*\). Moreover, the infimum in (1) is attained if and only if \(\lambda > \lambda^*\). The question whether the infimum is achieved in the critical case \(\lambda=\lambda^*\) remained open. The main result of the paper under review answers this question provided that the assumption \(p,q \in C^1(\overline{\Omega})\) is replaced by \(p,q \in C^2 (\overline{\Omega})\), and \(\eta \in C^0(\overline{\Omega})\) by \(\eta \in \text{Lip} (\overline{\Omega})\). This result reads as follows: If \(\lambda = \lambda^*\), then the infimum in (1) is attained if and only if \[ \int_{\partial\Omega} (1-q(\sigma)/p(\sigma))^{-1/2} d\sigma<\infty. \]
scientific article; zbMATH DE number 1423458